Oblique propagation of the squeezed states of s(p)-polarized light through non-Hermitian multilayered structures

Elnaz Pilehvar; Ehsan Amooghorban; Ehsan Amooghorban; Mohammad Kazem Moravvej-Farshi

doi:10.1364/OE.448229

1. Introduction

Wave scattering by non-Hermitian structures has attracted a great deal of attention over the past two decades. Particular interest is devoted to the special class of non-Hermitian media, referred to as parity-time ($\mathcal{PT}$-) symmetric in quantum mechanics. $\mathcal{PT}$-symmetric systems are described by a Hamiltonian that is invariant under parity-time symmetry transformation. In a mathematical language, this means that if the Hamiltonian $\hat{H}$ commutes with the ${\mathcal{\hat{P}}{\hat{T}}}$ operator, $[{\hat{H},{\mathcal{\hat{P}}{\hat{T}}}} ]= 0$, and they share the same set of eigenstates, then the eigenstates $\hat{H}$ are entirely real [1–3]. For optical $\mathcal{PT}$-symmetric systems, the refractive index obeys the symmetry relation n(r)=n*(−r) wherein r and asterisk denote the position and complex operators [4–7]. This possesses the perfectly balanced spatial distribution of gain and loss for physical systems. The given symmetry is the only necessary condition for the so-called “exact” phase or $\mathcal{PT}$-symmetric regime. The $\mathcal{PT}$-symmetry exact phase regime sustains when the gain/loss strength remains below a threshold, known as the exceptional point. Otherwise, the $\mathcal{PT}$-symmetry broken phase regime prevails, resulting in a complex eigenvalue spectrum [1–7].

Extension of $\mathcal{PT}$-symmetry to systems with eigenvalues that correspond to the scattering matrix has led to some novel phenomena: optical switching [8], nonreciprocal propagation [9,10], unidirectional reflectionless transmission [11,12], unidirectional invisibility [13–15], some extraordinary nonlinear effects [16–18], extraordinary transmission and reflection [19], optoelectronic oscillators [20], and phase transition in $\mathcal{PT}$-symmetric active plasmonic systems [21]. All these make use of classical electromagnetic waves. Despite the successful implementation of $\mathcal{PT}$-symmetry in the classical systems, some controversial results are proposed in the full-quantum regime, such as ultrafast states transformation and violation of the no-signaling principle [22,23]. However, investigations on the interaction of quantum waves and matters have left open the question of “whether the light retains its quantum features, after the interaction.”

Nonclassical states of light, such as squeezed states, can provide distinct properties, like reduced noise and strong correlations [24] compared with classical light and have found numerous applications in sensing and quantum metrology beyond the fundamental limits of precision set for classical light [25,26]. The quantum states of light propagating through absorbing or amplifying media are affected by quantum noise associated with the loss and gain [27–35]. Unlike in classical optics, so far, a few reports have explored the potential benefits of $\mathcal{PT}$-symmetric bilayers in quantum optics [35–40]. In our most recent work [35], we extensively studied the quantum states of light normally propagating through a dispersive non-Hermitian bilayer structure. There, we have demonstrated how the dispersion and the gain(loss)-induced noises in such structures affect the incident light quantum features, like squeezing and sub-Poissonian statistics. Then, we have demonstrated below a critical value of gain quantum optical effective medium theory (QOEMT) correctly predicts the behavior of quantized waves propagating through the non-Hermitian bilayer. We have also shown that unidirectional invisibility cannot be realized in quantum optics.

In nonclassical optics, the general case of non-normal incidence is of interest since the scattering patterns strongly depend on the angle of incidence. In addition, we know the propagating behavior of the s- and p-polarized waves differ qualitatively for oblique incidences [28]. Motivated by the results of [35], obtained from a one-dimensional analysis of a non-Hermitian bilayer under normal incidence, we aim at a more general investigation providing deeper insight into the physics of non-Hermitian periodic structures beyond the first quantization. In doing so, by investigating the properties of s(p)-polarized squeezed state of light obliquely propagating across a non-Hermitian multilayered structure, here, we evaluate the nonclassical features like quadrature squeezing and the photon counting statistics of the transmitted light for various loss coefficients and incident angles. We study the effects of the dispersion and the loss(gain)-induced noises on the squeezing and sub-Poissonian statistics for s(p)-polarization in the exact-phase and broken-phase regimes. This examination can serve as a meaningful tool to more accurately assess the realization of $\mathcal{PT}$-symmetry in quantum optics. Our results can have many applications, specifically, those based on manipulating the photonic spin Hall effect with nonclassical light [41], wave propagation, and polarization control, like polarization rotators [42] and polarization modulators [43].

2. Method

2.1 Multilayered structure

Consider a finite non-Hermitian multilayered (Fig. 1) composed of a few pairs of homogeneous gain/loss nanolayer of the same thickness (l) infinitely extended in the x-y plane. The structures with the total thickness of L = (N−1)l is surrounded by the vacuum along the z-direction at | z | > L/2. The background dielectric permittivity of gain (loss) medium, ε_g(l), follows the Lorentzian form [44],

(1)$${\varepsilon _{{g(l)}}}(\omega )= {\varepsilon _{{bg(l)}}} - \frac{{{\alpha _{{g(l)}}}\,{\omega _{{0g(l)}}}\;{\gamma _{{g(l)}}}}}{{{\omega ^2} - \omega _{{0}\,{g(l)}}^2 + i\omega \,{\gamma _{{g(l)}}}}},$$

in which ω is the incident light radian frequency, γ_g(l) and ω_0g(l) denote the emission (absorption) linewidth and the corresponding radian frequency, and α_g(_l) represents the nanolayers gain (loss) coefficient. Due to the causality principle, we consider α_g < 0 and γ_g > 0 (for the gain layer) and α_l > 0 and γ_l > 0 (for the loss layer). To satisfy the necessary condition for $\mathcal{PT}$-symmetry, we set ε_g(z, ω) = ε_l*(−z, ω). Notice that the $\mathcal{PT}$-symmetry condition for dispersive optical structures can be satisfied only for a discrete set of real frequencies [45].

Fig. 1. A 3D representation of a multilayered non-Hermitian structure consists of alternating gain and loss slabs with permittivities ε_g and ε_l, respectively, and equal thicknesses of l along the z-direction. The arrows with the annihilation operators show the input and output modes. The squeezing parameter of the outgoing light on the right-hand side of the structure is measured with a balanced homodyne detector.

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The plasmonic metamaterial with a lossless glass substrate wherein the quantum noise flux vanishes can provide an experimental proposal for this structure [46–48].

2.2 Exact multilayer theory

Assume an s(p)-polarized quantum state of light impinging at an angle of incidence, θ, from the left (right) upon the multilayered structure. According to the second quantization formalism of the electromagnetic field, the transverse positive Fourier component of the electric field operator in the j^th layer is [28]:

(2)$$\begin{aligned} {\hat{E}}_\sigma ^{ + (j)}({z,{\textbf k},\omega } )&= \frac{{i\omega }}{{{\beta _j}c}}\sqrt {\frac{{\hbar {{\beta ^{\prime}}_j}}}{{2{\varepsilon _0}}}} \{{\exp ({i{{\beta^{\prime}}_j}z} )\hat{a}_{R,\sigma }^{(j)}({z,{\textbf k},\omega } ){\textbf e}_{R,\sigma }^{(j)}({\textbf k})} \\ &+ {\exp ({ - i{{\beta^{\prime}}_j}z} )\hat{a}_{L,\sigma }^{(j)}({z,{\textbf k},\omega } ){\textbf e}_{L,\sigma }^{(j)}({\textbf k})} \}, \end{aligned}$$

where c and ε₀ represent the free space light velocity and permittivity, k and β_j (= ${\beta}^{\prime}_\text{j}+i{\beta}^{\prime\prime}_\text{j}$) are the in-plane and out-of-plane components of the wavevector in the layer, σ represents the wave polarization (s or p), R(L) denotes the right (left) propagation wave and ${\textbf e}_{R(L),\sigma }^{(j)}$ is the polarization vectors for s- and p-polarized waves propagating in the +(−) z-direction. The relation between the output amplitude operators, $\hat{a}_{L,\sigma }^{(1)}({{z_1}\,,{\textbf k},\omega \,} )$ and $\hat{a}_{R,\sigma }^{(N + 1)}({{z_{N}},{\textbf k},\omega \,} )$, at the left (z₁ = −L / 2) and right (z_N = +L / 2) boundaries and the operators of the incoming fields, $\hat{a}_{L,\sigma }^{(N + 1)}({{z_{N}},{\textbf k},\omega } )$ and $\hat{a}_{R,\sigma }^{(1)}({{z_1},{\textbf k},\omega } )$, at the opposite boundaries and the noise amplitude operators, ${\hat{F}_{R(L),\sigma }}({{\textbf k},\omega } )$, are as follows:

(3)$$\left( {\begin{array}{c} {\hat{a}_{L,\sigma }^{(1)}({{z_1},{k},\omega } )}\\ {\hat{a}_{R,\sigma }^{(N + 1)}({{z_N},{k},\omega } )} \end{array}} \right) = {\mathrm{\mathbb{S}}_\sigma }\left( {\begin{array}{c} {\hat{a}_{R,\sigma }^{(1)}({{z_1},{k},\omega } )}\\ {\hat{a}_{L,\sigma }^{(N + 1)}({{z_N},{k},\omega } )} \end{array}} \right) + \left( {\begin{array}{c} {{{\hat{F}}_{L,\sigma }}({{k},\omega } )}\\ {{{\hat{F}}_{R,\sigma }}({{k},\omega } )} \end{array}} \right)\,\,\,,\,\,\,\,{\mathrm{\mathbb{S}}_\sigma } \equiv \left( {\begin{array}{cc} {{r_{L,\sigma }}}&{{t_\sigma }}\\ {{t_\sigma }}&{{r_{R,,\sigma }}} \end{array}} \right),$$

wherein $ \mathbb{S} $_σ is a general scattering matrix, consisting a transmission coefficient, t_σ, which is the same for the right- and left-propagating modes at z = z_N and z₁ and two reflection coefficients, r_L,_σ, from the left boundary (at z₁) and r_R,_σ from the right boundary (at z_N) as in classical optics. The quantum noise, ${\hat{F}_{R(L),\sigma }},$ originating from the loss and gain layers, represents an internal source of light with no classical analog. Moreover, the optical input annihilation operators in (3) satisfy the bosonic commutation relations,

(4a)$$\begin{array}{c} [{\hat{a}_{R,\sigma }^{(1)}({z,{k},\omega } ),\hat{a}_{R,\sigma^{\prime}}^{(1)\dagger }({z^{\prime},{k^{\prime}},\omega^{\prime}} )} ]= [{\hat{a}_{L,\sigma }^{(N + 1)}({z,{k},\omega } ),\hat{a}_{L,\sigma^{\prime}}^{(N + 1)\dagger }({z^{\prime},{k^{\prime}},\omega^{\prime}} )} ]\\ = {\delta _{\sigma \sigma ^{\prime}}}\delta ({\omega - \omega^{\prime}} )\delta ({{k} - {k^{\prime}}} ), \end{array}$$

(4b)$$[{\hat{a}_{R,\sigma }^{(1)}({z,{k},\omega } ),\hat{a}_{L,\sigma^{\prime}}^{(N + 1)\dagger }({z^{\prime},{k^{\prime}},\omega^{\prime}} )} ]= 0.$$

The explicit forms of ${\hat{F}_{R(L),\sigma }}$ and the scattering matrix elements can be found in [28]. Similar commutation relations for the output amplitude operators can be derived by substituting (3) in (4).

3. Results and discussion

In our investigations, we have considered two sets of unit-cells, with physical parameters given in Table 1 to constitute the multilayered structure of Fig. 1. These parameters are the same as those used in [44,49]. As listed in this table, the gain and loss layers with the same thickness (l = 10 nm) for Set 1 (Set 2) have identical (unidentical) background permittivities. In other words, Δε = ε_bl − ε_bg = 0 (1.22) for Set 1 (Set 2).

Table 1. Physical parameters for two sets of unit-cells, constituting the multilayered structure of Fig. 1, used in the simulations [44,49].

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According to Eq. (1) and Table 1, the frequency, the loss, and the gain parameters that satisfy the $\mathcal{PT}$-symmetric necessary condition for Set 1(Set 2) are $\omega = \omega_{\mathcal{PT}} = \omega_{\rm 0g}$ (1.58 ω_0g) for any arbitrary value of α_l =| α_g | (solely for α_l= 2 and α_g=−20.86). We use the given parameters to investigate the behavior of the outgoing optical beam at z = z_N (z₁) when an s(p)-polarized quantum state obliquely impinges from the left (right) upon the slab z = z₁ (z_N).

3.1 Scattering matrix eigenvalues

In general, for both polarizations, the eigenvalues of the scattering matrix (3) for the exact $\mathcal{PT}$-symmetric phase are unimodular, fulfilling the condition |λ_1,_σ λ_2,_σ| = 1. Hence, up to the so-called exceptional-point we have |λ_1,_σ| = |λ_2,_σ| = 1, beyond which the broken phase regime emerges, and the eigenvalues become an inverse conjugate pair, satisfying the relation $|\lambda_1, \sigma|$=| λ_2,_σ |^-1 > 1 (i.e., the so-called $\mathcal{PT}$-broken phase) [5,11]. From Eq. (3), we can write the eigenvalues for a unit-cell (a bilayer):

(5)$${\lambda _{1,2,\sigma }} = \frac{1}{2}({{r_{L,\sigma }} + {r_{R,\sigma }}} )\pm \frac{1}{2}\sqrt {{{({{r_{L,\sigma }} - {r_{R,\sigma }}} )}^2} + 4t_\sigma ^2} .$$

To identify whether the system lies in the $\mathcal{PT}$-symmetric or $\mathcal{PT}$-broken phase, we examine the dependency of the eigenvalues difference, Δλ_σ= | λ_2,σ | − | λ_1,σ |, on the loss coefficient (α_l) and the incident angle (θ) for both polarizations. We assumed an s(p)-polarized quantum state impinging upon the structure of Fig. 1 composed of Set 1 unit-cell at an arbitrary angle θ and calculated the corresponding eigenvalues versus θ and α_l. Figures 2(a) and 2(b) illustrated Δλ_σ in the (θ, α_l) plane for s- and p-polarization. As shown in this figure, for any given value of 0 ≤ θ ≤ 85° and α_l < 890, the $\mathcal{PT}$-symmetric exact phase regime for both polarizations prevail (i.e., Δλ_σ = 0). Moreover, the exceptional point occurs at α_l = 890 (marked with vertical dashes), beyond which the so-called broken phase occurs for both polarizations at any given angle of incidence. This case is contrary to that reported for the chiral $\mathcal{PT}$-symmetric materials [50–52], in which s- and p-polarizations showed different exceptional points at any given incident angle except θ = 0°. Figure 2(c) and 2(d), illustrating λ_1,2(σ) versus α_l for θ = 30° and 60° manifest the nature of the eigenvalues in each regime. Being unimodular in the exact phase regime and nonunimodular in the broken phase regime. Further comparison shows, unlike the exact-phase eigenvalues, those of the broken-phase depend on the incident angle and polarization.

Fig. 2. (a) and (b) The profiles of differences in the eigenvalues (Δλ_σ= | λ_2,σ | − | λ_1,σ |) versus θ and α_l, for s- and p-polarizations (c) and (d) | λ_1,2,σ | versus α_l at θ = 30° and 60°; all for the unit-cell of Set 1. The vertical dashes denote the exceptional point in each case.

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As we have pointed out earlier in this section, the Set 2 unit-cell only at α_l = 2 (| α_g | = 20.86), the condition | λ₁,_σ | = | λ₂,_σ | = 1 is satisfied. Otherwise, this unit-cell cannot be $\mathcal{PT}$-symmetric for any given angle of incidence and polarization (see [35]). In the following subsections, we investigate how quantum features of the incident state (here, the squeezing and sub-Poissonian statistics) are degraded in the $\mathcal{PT}$-symmetry and also in the broken phase regimes for Set 1 and Set 2.

3.2 Quadrature squeezing

In this section, we study how propagation through the multilayered structure influences the squeezing of the incident light. In doing so, we take the state of the quantized field |L〉=|0〉 (a conventional vacuum) impinging leftward and |R〉 (a squeezed coherent continuous-mode) impinging rightward upon the multilayered structure. The balanced homodyne detector (shown on the right-hand side of Fig. 1) can measure the quadrature squeezing of the scattered light. In other words, a 50-50 beam splitter mixes the outgoing light from the structure with the coherent mode of the local oscillator (LO) of frequency ω_LO and phase ϕ_LO. Consider a narrow-bandwidth homodyne detector is functioning during a sufficiently long-time interval [27–31]. After some lengthy mathematical manipulations, one can obtain a compact formula for the variance of the difference between the photocurrents given by [28,31],

(6)$$\begin{aligned} {\left\langle {{{({\varDelta \hat{E}_\sigma({\phi_{LO}},{\textbf k},{\omega_{LO}})} )}^2}} \right\rangle ^{out}} &= 1 + 2\left\langle {\hat{F}_{R,\sigma }^\dagger ({{\omega_{LO}},{\textbf k}} ){{\hat{F}}_{R,\sigma }}({{\omega_{LO}},{\textbf k}} )} \right\rangle \\ &+ {|{{t_\sigma }({\textbf k},{\omega_{LO}})} |^2}\{{2{{\sinh }^2}{\xi_\sigma } - Re ({e^{2i({\phi_{\textrm{LO}}} - {\phi_t} - {\phi_{\xi ,\sigma }}/2)}}\sinh 2{\xi_\sigma })} \}\\ &+ {|{{r_{R,\sigma }}({\textbf k},{\omega_{LO}})} |^2}\{{2{{\sinh }^2}{{\xi^{\prime}}_\sigma } - Re ({e^{2i({\phi_{\textrm{LO}}} - {\phi_R} - {\phi_{\xi^{\prime},\sigma }}/2)}}\sinh 2{{\xi^{\prime}}_\sigma })} \}, \end{aligned}$$

where ξ represents the squeezed parameter that controls the degree of squeezing and its corresponding phase, ϕ_ξσ, which depends on the frequency, polarization, and the angle of incidence (see Eqs. (S1)–(S11) in the Supplement 1 for the derivation steps leading to Eq. (6)). Moreover, ϕ_t and ϕ_R are the phases of the transmission and right reflection coefficients. The details of the average flux of the noise photons, representing the loss and gain within the structure —i.e., $\left\langle {\hat{F}_{R,\sigma }^\dagger ({{\omega_{LO}},{\textbf k}} ){{\hat{F}}_{R,\sigma }}({{\omega_{LO}},{\textbf k}} )} \right\rangle$— are given by Eq. (53) of Ref. [28]. For the vacuum quantum state in the absence of the structure 〈(ΔE)²〉^out = 1, and squeezing occurs when one of the quadrature components of the scattered light drops below the vacuum level—i.e., 〈(ΔE)²〉^out < 1. In our calculations, we take the parameters ξ_σ= 0.2 and ϕ_ξσ= 2ϕ_LO – 5 for the squeezed coherent state |R〉 and equivalent meaning of ξ_σ for the vacuum state |L〉 to be ξ′_σ=0 [31]. Making use of the exact multilayer theory, we calculate the variance 〈(Δ$\hat{E}$)²〉_σ^out for the states transmitted through the $\mathcal{PT}$-symmetric structure of Fig. 1 versus α_l and θ. Figure 3 illustrates the output quadrature variance of the (i) s- and (ii) p-polarizations from two $\mathcal{PT}$-symmetric structures composed of (a) one and (b) four unit-cells designated by Set 1 at frequency ω_LO = ω_0g and zero temperature (0 K), in the α_l−θ plane. For the sake of clarity, we show plots of 〈(Δ$\hat{E}$)²〉_σ^out for the structure composed of four unit-cells for both polarizations, at θ = 0, 30, and 60° in Fig. 3(c). As can be seen from Fig. 3, always 〈(Δ$\hat{E}$)²〉_σ^out > 1, the transmitted states through both given structures are no longer squeezed over the entire α_l−θ plane for both polarizations. This effect is attributed to the contribution of the quantum noise, at 0 K, originating from the gain nanolayers. In other words, at 0 K, the perfect population inversion at ω_0g maximizes the noise flux via the spontaneous emission of the atoms, significant enough to eliminate the squeezing property of the incident state. Notice, for large values of α_l, these two structures perform as partial mirrors (see Fig. S1 in Supplement 1 for one unit-cell) for both polarizations. Moreover, at α_l = 1000, all variances exceed unity slightly.

Fig. 3. Profile of ($\langle(\Delta\hat{E})^2\rangle$ _σ^out in the α_l-θ plane for (i) s-polarization and (ii) p-polarization transmitted through the structure made of (a) one and (b) four unit-cells designated by Set 1. (c) The data extracted from part (b) for s- (magenta) and p (blue)-polarized lights, at θ = 0° (dashes), 30°(dots), and 60°(line). Notice, ω_LO = ω_0g.

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Next, we have repeated similar calculations for non-Hermitian structures, made of (a) one and (b) four unit-cells designated by Set 2 (see Fig. 4). As seen from this figure, there are specific regions in the α_l−θ plane over which the transmitted states through these non-Hermitian structures for both polarizations remained squeezed. A careful comparison of the data in Figs. 4(a) and 4(b) reveal that as the number of the constituting unit-cells in these multilayered structures increases, the area of the corresponding squeezed region decreases. This phenomenon is due to the significant increase in the quantum noises within the structure originating from the gain layers at 0 K. Unlike for Set 1, the quantum noise for Set 2 at the incident frequency of ω_LO =1.58 ω_0g is negligible, having little effect on the squeezing property of the incident state. In other words, the transmitted states through these multilayered structures, to some extent, retain the squeezing property of the incident states.

Fig. 4. Profile of 〈(Δ$\hat{E}$)²〉_σ^out in the α_l-θ plane for (i) s-polarization and (ii) p-polarization transmitted through the structure made of (a) one and (b) four unit-cells designated by Set 2 (c) The data extracted from part (b) for s- (magenta) and p (blue)-polarized lights, at θ = 0° (dashes), 30°(dots), and 60°(line). Notice, ω_LO = 1.58ω_0g, and the regions in (a) and (b) surrounded by dots represent the squeezed region, so does the light green region in (c).

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The regions in Fig. 4(a) and 4(b) enclosed by the green dots represent the squeezed region. Notice, these non-Hermitian structures behave like lossless (gainless) structures with negligible reflectances (R_R ≈ R_L < 30%) and large transmittances (T > 80%) for small values of α_l and the given range of θ (see Fig. S2 in Supplement 1). This property makes the noise flux vanishingly small, which adds up with the small positive values of the transmission-related term (i.e., the third term on the right-hand side in Eq. (6)), contributing to the squeezing variance to make it slightly greater than one. A moderate increase in α_l, results in a dominantly negative transmitted-related term due to the presence of the α_l-dependent phase, ϕ_t, resulting in 〈(Δ$\hat{E}$)²〉_σ^out < 1. As α_l exceeds a particular value, at any given θ, each of these two non-Hermitian structures behaves like a partial mirror with R_R ≈ R_L > 80% and T < 20% for both polarizations (see Fig. S2, in Supplement 1). This behavior forces the scattering mode to approach the vacuum state with the variance of 〈(Δ$\hat{E}$)²〉_σ^out ≈ 1, at large α_l values.

Comparing the six plots in Figs. 3(c) or 4(c), we realize that the magenta and blue dashes (representing θ = 0 for s- and p- polarizations) in either figure coincide over the entire range of the given α_l. In other words, the light polarization does not affect the propagation of the normally incident quantum states passing through either of the two sets of non-Hermitian multilayered structures (Set 1 and Set 2), as expected. Nonetheless, the deviation of magenta dotted and solid lines from their blue counterparts in each figure indicate that the former statement is not valid for oblique incidences (i.e., θ > 0). Moreover, we can see as the θ increases, the variance spectrum for the s(p)-polarization compresses (expands) towards smaller (larger) α_l, with a larger (smaller) minimum.

It is worth noting that the incident and transmitted light through the non-Hermitian structure (constituting loss/gain unit-cells) designated as Set 1 are not equivalent. Despite the apparent simplicity of our example, it does have far-reaching consequences for any attempt to study $\mathcal{PT}$-symmetry of quantum optical systems. First of all, the gain layer, regardless of its spatial distribution, always adds thermal noise to a quantum state at zero temperature. Second, we have chosen squeezed states that are minimum uncertainty states while having the unique property, such that loss cannot affect their purity. For example, pure photon-number states turn into mixed states with lower photon numbers. Hence, any quantum state of light is crucially altered when propagating through Set 1 structures. In other words, Hamiltonians for such non-Hermitian structures do not commute with the $\mathcal{PT}$ operator (i.e., $[{\hat{H},\hat{P}\hat{T}} ]\ne 0$); and the nonclassical features of transmitted light are entirely lost. Nonetheless, the quantum states of light after transmitting through the non-Hermitian structure of Set 2 can preserve their nonclassical features to some extent for some given values of α_l and θ. This is an apparent manifestation of broken $\mathcal{PT}$ -symmetry in quantum optics as far as the squeezing feature of the outgoing light is concerned.

3.3 Photon counting statistics

The so-called Mandel parameter is a practical tool for characterizing the photon counting statistics in the transmitted nonclassical light field through the multilayered structure [31]. Using a photo count detector with a Gaussian filter of bandwidth σ_H and center frequency ω_H, in which ω_H ≫ σ_H, one can measure the photon counting statistics. Moreover, after some lengthy manipulations, the Mandel parameter becomes

(7)$$\begin{aligned} {Q_\sigma }({\textbf k},\omega ) &= {Q_0}\{{{{({1 - {{|{{r_{R,\sigma }}({\textbf k},\omega )} |}^2}} )}^2} + {{|{t_\sigma^{}({\textbf k},\omega )} |}^4}[{1 + {{\sinh }^2}{\xi_\sigma }({\cosh 2{\xi_\sigma } - 2} )} } \\ &+ 2{\sigma _H}\sqrt \pi {|\rho |^2} {({2{{\sinh }^2}{\xi_\sigma } + \sinh 2{\xi_\sigma }\cos ({2{\varphi_\rho } - {\varphi_{\xi ,\sigma }}} )- 2} )} ]\\ &\left. { + 2{{|{t_\sigma^{}({\textbf k},\omega )} |}^2}({1 - {{|{{r_{R,\sigma }}({\textbf k},\omega )} |}^2}} )\left( {{{\sinh }^2}{\xi_\sigma } + 2{\sigma_H}\sqrt \pi {{|\rho |}^2} - 1} \right)} \right\}\\ &{\left\{ {{{|{t_\sigma^{}({\textbf k},\omega )} |}^2}\left( {{{\sinh }^2}{\xi_\sigma } + 2{\sigma_H}\sqrt \pi {{|\rho |}^2}} \right) + \left\langle {\hat{F}_{R,\sigma \,}^\dagger ({{\textbf k},\omega } )\hat{F}_{R,\sigma \,}^{}({{\textbf k^{\prime}},\omega^{\prime}} )} \right\rangle } \right\}^{ - 1}}, \end{aligned}$$

where Q₀= σ_HT₀/4π^3/2. The Mandel parameter for the transmitted light, in the range of Q_σ < 0, Q_σ = 0, or Q_σ > 0, corresponds to the so-called sub-Poissonian (a quantum feature of the optical field), Poissonian, or super-Poissonian photon distribution, respectively. Considering ϕ_ξ= (2ϕ_ρ−π) and 2σ_H π^1/2 | ρ |²= 25 with Q_i / Q₀= −0.33 as the normalized Mandel parameter for the incident squeezed coherent state |R〉 (similar to that used in [28,31]), we show the effects of the dispersion and loss(gain)-induced noises on the photon counting statistics by the incident states same as the previous subsection.

Figures 5(a) and 5(b) illustrate the profiles of the Mandel parameter (Q_σ / Q₀) in the α_l-θ plane obtained at 0 K for the Set 1 made of (a) one and (b) four unit-cells for the (i) s-polarized and (ii) p-polarized output states. As shown in Fig. 5(a) and 5(b), a vast area in the α_l-θ plane corresponds to the super-Poissonian statistics for s- and p-polarizations. That is because of the significant quantum noises within this set of the $\mathcal{PT}$-symmetric structures at ω = ω_0g and the given range of α_l−θ. Moreover, the Mandel parameter distribution for each case (i or ii) may vary from one polarization to another. One can attribute this variation to the differences in the corresponding transmittances and the right (left) reflectance for the obliquely incident s-and p-polarized lights (see Fig. S1 in Supplement 1). For example, in a particular area of the α_l−θ plane, the profile in Fig. 5(a-i or b-i) may represent sub-Poissonian statistics. While, in a similar region, Fig. 5(a-ii or b-ii) may represent super-Poissonian statistics and vice versa. This dependency demonstrates the possibility of controlling the quantum features of the transmitted light through non-Hermitian systems by polarization. The green dots in Fig. 5(a) and 5(b) represent the boundaries between sub-Poissonian and super-Poissonian regions —i.e., the loci of Q_σ = 0. A further inspection shows, an increase in the number of unit-cells constituting the structure expands the area covered by super-Poissonian distribution for both s- and p-polarized states. This phenomenon is due to the significant quantum noises within the Set 1 structures originating from the gain layers at 0 K.

Fig. 5. Profile of the Mandel parameter Q/Q₀ in the α_l-θ plane for (i) s-polarization and (ii) p-polarization transmitted through the structure made of (a) one and (b) four unit-cells designated by Set 1. The regions surrounded by green dots represent the sub-Poissonian statistics (S-P). (ci) The data extracted from part (b) for s (magenta) and p (blue)-polarized lights, at θ = 0° (dashes), 30°(dots), and 60°(line). The inset in (c) shows a zoomed-in portion of the plots for small values of α_l where the output states are sub-Poissonian, i.e., the light green region. Notice, ω_LO = ω_0g.

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Again, we focus on three particular angles — i.e., at θ = 0°, 30°, and 60° — and extract data from Figs. 5(b) to further compare the dependency of Q_σ / Q₀ on the loss coefficient, α_l, at the given angles of incidence for the structures made of four unit-cells of Set 1, as depicted in Fig. 5(c). The inset in this figure shows only for small values of α_l the photon statistics are sub-Poissonian (Q_σ / Q₀ <0), while the squeezing character is not retained (see Fig. 3 (c)). Here too, alike in Fig. 3(c), the magenta and blue dashes coincide over the entire range of the given α_l, meaning the propagation of normally incident (θ = 0°) quantum states through a planar structure are polarization independent. Recall that for large values of α_l, this set of structures multilayers perform as a mirror for both polarizations. Albeit, the Mandel parameter for any given angle of incidence is slightly greater than zero at α_l= 1000.

Now, we move to Set 2, investigating the variations of the Mandel parameter, similar to those shown in Fig. 5 for Set 1. Figure 6 shows these variations versus α_l and θ for the s- and p- polarized states transmitted through Set 2 non-Hermitian structures made of (a) one and (b) four unit-cells at ω_LO = 1.58ω_0g. As can be seen from this figure, this set of non-Hermitian slabs, unlike Set 1, preserves the sub-Poissonian statistics of the incident states for both polarizations, over the entire given α_l-θ plane. The dots in Fig. 6(a) and 6(b) represent the boundaries between sub-Poissonian and Super-Poissonian statistics — i.e., the loci of Q_σ=0. This effect can be seen more clearly from plots of Fig. 6(c), depicting the dependence of the Mandel parameters on the loss coefficient, at θ = 0°, 30°, and 60°. One can attribute this phenomenon to the incident frequency being far away from the emission frequency of the gain layers in the Set 2 structure, making the quantum noise infinitesimal, even at α_l = 1000. Moreover, the difference between profiles of the Mandel parameters for different polarizations corresponds to the difference in their related transmittances and right and left reflectances in the similar range of α_l,observed in Fig. S2 for one unit cell. Recall the Set 2 structure behaves like a lossless (gainless) structure with negligible reflectances and large transmittances for small loss values. Nonetheless, it acts as a mirror at the large values of α_l. The latter behavior forces the scattering mode to approach the coherent state with a zero Mandel parameter for large loss values. The results of Fig. 6(c), as far as the dependence of the Mandel parameter on the incident polarization and angle is solely concerned, show that the structure composed of Set 2 is a suitable candidate to realize the $\mathcal{PT}$-symmetry in quantum optics.

Fig. 6. Profile of the output Mandel parameter Q_σ/Q₀ in the α_l-θ plane for (i) s-polarization and (ii) p-polarization transmitted through the structure made of (a) one and (b) four unit-cells designated by Set 2. The dots in parts (b) represent the boundaries between super-Poissonian and sub-Poissonian regions. (c) The data extracted from part (b) for s (magenta) and p (blue)-polarized lights, at θ = 0° (dashes), 30°(dots), and 60°(line). The region shaded light green in this part represents the sub-Poissonian. Notice, ω_LO = 1.58ω_0g.

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4. Conclusions

We have investigated the behavior of obliquely incident s- and p-polarized quantum states after transmitting through dispersive non-Hermitian media. To account for the dispersive, dissipative, and amplification properties, we employed the Lorentz model. Then, considering a squeezed coherent state of light with an arbitrary angle of incidence, we have investigated to see to what extent the transmitted light could retain its original nonclassical features, like the quadrature squeezing and sub-Poissonian photon statistics. In doing so, we have considered two sets of non-Hermitian structures: one with gain/loss layers of identical background materials (Set 1) and the other with unidentical background materials (Set 2). The frequencies of the incident squeezed coherent states were equal to the frequencies satisfying the $\mathcal{PT}$-symmetric necessary condition for both sets, which equals the emission frequency of gain layers in Set 1 (ω_LO = ω_0g) and is far away from Set 2 (ω_LO = 1.58ω_0g). Our findings for one unit-cell show that Set 2 can retain the sub-Poissonian photon statistics (squeezing) of the incident squeezed coherent states, to some extent over the entire (some parts of) α_l-θ plane for both s- and p-polarizations. However, Set 1 cannot retain the squeezing (sub-Poissonian photon statistics) of the incident squeezed coherent states over the entire (large parts of) α_l-θ plane. Hence, one cannot implement $\mathcal{PT}$-symmetry, at any arbitrary angle of incidence for either polarization in the quantum optics domain as far as the squeezing feature of outgoing light is concerned. Although, this situation is changed if one only probes the sub-Poissonian photon statistics of outgoing light, and it seems that the structure composed of Set 2 is a good choice. We have also found that for both polarizations, the quantum features of the transmitted state degrade as the number of unit-cells constituting the structures and the incident angle θ are increased. Moreover, the transmitted quantum states of light through these non-Hermitian multilayers have not behaved extraordinarily at exceptional points.

Funding

Tarbiat Modares University (IG-39703).

Acknowledgments

This work was supported by Tarbiat Modares University (TMU) (IG-39703).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

2. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT–symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999). [CrossRef]

3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]

4. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

5. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of $\mathcal{PT}$-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef]

6. K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Beam dynamics in $\mathcal{PT}$-symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef]

7. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical $\mathcal{PT}$-symmetric structures,” Opt. Lett. 32(17), 2632 (2007). [CrossRef]

8. F. Nazari, M. Nazari, and M. K. Moravvej-Farshi, “A 2×2 spatial optical switch based on PT–symmetry,” Opt. Lett. 36(22), 4368–4370 (2011). [CrossRef]

9. M. Nazari, F. Nazari, and M. K. Moravvej-Farshi, “Dynamic behavior of spatial solitons propagating along Scarf II parity-time symmetric cells,” J. Opt. Soc. Am. B 29(11), 3057 (2012). [CrossRef]

10. F. Nazari, M. Nazari, M. K. Moravvej-Farshi, and H. Ramezani, “Asymmetric evolution of interacting solitons in parity time-symmetric cells,” IEEE J. Quantum Electron. 49(11), 932–938 (2013). [CrossRef]

11. L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional $\mathcal{PT}$-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). [CrossRef]

12. S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through $\mathcal{PT}$-symmetric epsilon-near-zero-bilayers,” Phys. Rev. B 89(8), 085105 (2014). [CrossRef]

13. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]

14. Y. Shen, X. Hua Deng, and L. Chen, “Unidirectional invisibility in a two-layer non-$\mathcal{PT}$-symmetric slab,” Opt. Express 22(16), 19440 (2014). [CrossRef]

15. M. Sarisaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a $\mathcal{PT}$-symmetric slab system,” Phys. Rev. A 95(1), 013806 (2017). [CrossRef]

16. F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via $\mathcal{PT}$-symmetric nonlinear Fano resonances,” Opt. Express 22(8), 9574–9584 (2014). [CrossRef]

17. F. Nazari and M. K. Moravvej-Farshi, “Multi-channel optical isolator based on nonlinear triangular parity time-symmetric lattice,” IEEE J. Quantum Electron. 52(8), 1–7 (2016). [CrossRef]

18. S. Xia, D. Kaltsas, D. Song, I. Komis, J. Xu, A. Szameit, H. Buljan, K. G. Makris, and Z. Chen, “Nonlinear control of $\mathcal{PT}$-symmetry and non-Hermitian topological states,” Science 372(6537), 72–76 (2021). [CrossRef]

19. J. Wu and X. Yang, “Ultrastrong extraordinary transmission and reflection in $\mathcal{PT}$-symmetric Thue-Morse optical waveguide networks,” Opt. Express 25(22), 27724 (2017). [CrossRef]

20. L. Li, Y. Cao, Y. Zhi, J. Zhang, Y. Zou, X. Feng, B. Guan, and J. Yao, “Polarimetric parity-time symmetry in a photonic system,” Light: Sci. Appl. 9(1), 169 (2020). [CrossRef]

21. M. Mattheakis, T. Oikonomou, M. I. Molina, and G. P. Tsironis, “Phase transition in $\mathcal{PT}$-symmetric active plasmonic systems,” IEEE J. Sel. Top. Quantum Electron. 22(5), 76–81 (2016). [CrossRef]

22. Y. Lee, M. Hsieh, S. T. Flammia, and R. Lee, “Local PT symmetry violates the no-signaling principle,” Phys. Rev. Lett. 112(13), 130404 (2014). [CrossRef]

23. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007). [CrossRef]

24. I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348(6234), 525–530 (2015). [CrossRef]

25. The LIGO Scientific Collaboration, “Enhancing the sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7(8), 613–619 (2013). [CrossRef]

26. S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, “Quantum amplification of mechanical oscillator motion,” Science 364(6446), 1163–1165 (2019). [CrossRef]

27. E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110(15), 153602 (2013). [CrossRef]

28. E. Amooghorban and M. Wubs, “Quantum optical effective-medium theory for layered metamaterials,” Arxiv:1606.07912v1, (2016).

29. E. Pilehvar, E. Amooghorban, and M. K. Moravvej-Farshi, “Quantum squeezed light propagation in an optical parity-time (PT)-symmetric structure,” Int. J. Optics and Photonics (IJOP 13(2), 181–188 (2020). [CrossRef]

30. R. Matloob and G. Pooseh, “Scattering of coherent light by a dielectric slab,” Opt. Commun. 181(1-3), 109–122 (2000). [CrossRef]

31. M. Artoni and R. Loudon, “Propagation of nonclassical light through an absorbing and dispersive slab,” Phys. Rev. A 59(3), 2279–2290 (1999). [CrossRef]

32. R. Asadi Aghbolaghi, E. Amooghorban, and A. Mahdifar, “Scattering of sphere coherent state by an absorptive and dispersive dielectric slab,” Eur. Phys. J. D 71(11), 272 (2017). [CrossRef]

33. M. Hoseinzadeh, E. Amooghorban, A. Mahdifar, and M. A. Nafchi, “Quantum light propagation in a uniformly moving dissipative slab,” Phys. Rev. A 101(4), 043817 (2020). [CrossRef]

34. H. Schomerus, “Quantum noise and self-sustained radiation of $\mathcal{PT}$-symmetric systems,” Phys. Rev. Lett. 104(23), 233601 (2010). [CrossRef]

35. E. Pilehvar, E. Amooghorban, and M. K. Moravvej-Farshi, “Quantum optical analysis of squeezed state of light through dispersive non-Hermitian optical bilayers,” J. Opt. 24(2), 025201 (2022). [CrossRef]

36. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in the transmission of quantum fields in $\mathcal{PT}$-symmetric optical systems,” Phys. Rev. A 85(3), 031802 (2012). [CrossRef]

37. S. Scheel and A. Szameit, “$\mathcal{PT}$-symmetric photonic quantum systems with gain and loss do not exist,” Euro Phys. Lett. 122(3), 34001 (2018). [CrossRef]

38. F. Klauck, L. Teuber, M. Ornigotti, M. Heinrich, S. Scheel, and A. Szameit, “Observation of $\mathcal{PT}$-symmetric quantum interference,” Nat. Photonics 13(12), 883–887 (2019). [CrossRef]

39. J. Perina, A. Luks, J. K. Kalaga, W. Leonski, and A. Miranowicz, “Nonclassical light at exceptional points of a quantum $\mathcal{PT}$-symmetric two-mode system,” Phys. Rev. A 100(5), 053820 (2019). [CrossRef]

40. E. Pilehvar, E. Amooghorban, and M. K. Moravvej-Farshi, “Propagation of Quantum Squeezed Radiation in Symmetric Rydberg Atomic Structures,” Presented at 27^th Iranian Conf, on Electrical Engineering (ICEE2019) Yazd (Iran, 30 April-2 May) p 384 (2019). DOI: 10.1109/IranianCEE.2019.8786406.

41. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015). [CrossRef]

42. B. Meier, “Quantum-rotor-induced polarizations,” Magn. Reson. Chem. 56(7), 610–618 (2018). [CrossRef]

43. Y. Li, Y.-H. Li, H.-B. Xie, Z.-P. Li, W.-Q. Cai, J.-G. Ren, J. Yin, S.-K. Liao, and C.-Z. Peng, “High-speed robust polarization modulation for quantum key distribution,” Opt. Lett. 44(21), 5262–5265 (2019). [CrossRef]

44. O. V. Shramkova, K. G. Makris, D. N. Christodoulides, and G. P. Tsironis, “Dispersive non-Hermitian optical heterostructures,” Photonics Res. 6(4), A1 (2018). [CrossRef]

45. A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Liyansky, “Causality and phase transitions in $\mathcal{PT}$-symmetric optical systems,” Phys. Rev. A 89(3), 033808 (2014). [CrossRef]

46. A. Fang, T. Koschny, and C. M. Soukoulis, “Self-consistent calculations of loss-compensated fishnet metamaterials,” Phys. Rev. B 82(12), 121102 (2010). [CrossRef]

47. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Freestanding optical negative-index metamaterials of green light,” Nature 466(7307), 735–738 (2010). [CrossRef]

48. K. Tanaka, E. Plum, J. Y. Ou, T. Uchino, and N. I. Zheludev, “Multifold enhancement of quantum dot luminescene in plasmonic metamaterials,” Phys. Rev. Lett. 105(22), 227403 (2010). [CrossRef]

49. A. Fang, T. Koschny, and C. M. Soukoulis, “Lasing in metamaterials nanostructures,” J. Opt. 12(2), 024013 (2010). [CrossRef]

50. S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Accessible phases via wave impedance engineering with $\mathcal{PT}$-symmetric metamaterial,” Phys. Rev. B 100(20), 205133 (2019). [CrossRef]

51. I. Katsantonis, S. Droulias, C. M. Soukoulis, E. N. Economou, and M. Kafesaki, “$\mathcal{PT}$-symmetric chiral metamaterials: Asymmetric effects and PT-phase control,” Phys. Rev. B 101(21), 214109 (2020). [CrossRef]

52. S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Chiral metamaterials with $\mathcal{PT}$-symmetry and beyond,” Phys. Rev. Lett. 122(21), 213201 (2019). [CrossRef]

Symbol	Definition	size		unit
Symbol	Definition	Set 1	Set 2	unit
ε_bl	Background dielectric constant of the loss layers	2	3.22	−
ε_bg	Background dielectric constant of the gain layers	2	2	−
ε_l	Absorption linewidth of the loss layers	6.7×10¹³	1.4×10¹⁴	rad·s⁻¹
ε_g	Emission linewidth of the gain laye.rs	6.7×10¹³	67×10¹³	rad·s⁻¹
ω_0l	Absorption frequency of the loss layers	10¹⁵	1.2×10¹⁵	rad·s⁻¹
ω_0g	Emission frequency of the gain layers	10¹⁵	10¹⁵	rad·s⁻¹

Oblique propagation of the squeezed states of s(p)-polarized light through non-Hermitian multilayered structures

Abstract

1. Introduction

2. Method

2.1 Multilayered structure

2.2 Exact multilayer theory

3. Results and discussion

3.1 Scattering matrix eigenvalues

3.2 Quadrature squeezing

3.3 Photon counting statistics

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (6)

Tables (1)

Equations (8)

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